No restrictions on white?
The main constraint here is that Black has to get his knight to h1. This takes 4 moves.
Meanwhile Black moves no pawns and no pieces besides his knight. White has a quick mate by putting a queen or bishop on f2, supported by the other (bishop on c4 or queen on f3 or h5). This takes 4 moves (e.g. e4, Bc4, Qf3, Bxf7#) - which is too quick for the black knight to finish his journey.
White has to make an extra move somewhere (could be most anything) and can mate on his 5th move.
There are additional variations where that extra move is put to use. White's supporting piece could be the king's knight, or the king's pawn, either of which needs the time to get into position.
Inspired by this famous problem that stumped Kasparov, I thought up a similar, much easier puzzle. My apologies if this has already been posted!
Rules for the game:
1. Black's pieces must end in their original position OR be off the board, EXCEPT;
2. Black's g8 knight must end on h1.
Find the fastest mate for white!